A bipartite graph is a type of graph in which the vertices can be divided into two disjoint sets such that all edges connect vertices from different sets. In other words, there are no edges that connect vertices within the same set.

Explanation:

- Bipartite graph definition:
- A bipartite graph G = (V, E) is a graph whose vertex set V can be partitioned into two sets V1 and V2 such that every edge in E connects a vertex from V1 to a vertex from V2.

- Example of a bipartite graph:
- A simple example of a bipartite graph is a graph representing the relationship between students and classes. Let's say we have a set of students (V1) and a set of classes (V2). An edge between a student and a class indicates that the student is enrolled in that class. In this case, all the edges will connect a student from V1 to a class from V2, satisfying the definition of a bipartite graph.

- No cycles of odd length:
- One important property of bipartite graphs is that they do not contain any cycles of odd length. This can be proven by contradiction. Assume we have a bipartite graph with a cycle of odd length. Since the cycle is of odd length, the number of edges in the cycle is also odd. But according to the definition of a bipartite graph, all edges connect vertices from different sets. Therefore, the number of edges in the cycle must be even, which contradicts our assumption.

- Why no cycles of odd length?
- The absence of cycles of odd length in bipartite graphs is related to the concept of colorability. Bipartite graphs can be colored with two colors such that no adjacent vertices have the same color. This is known as a 2-coloring. If a cycle of odd length exists in a bipartite graph, it would require three colors to color the vertices in the cycle without adjacent vertices having the same color. But since a bipartite graph can be 2-colored, the existence of a cycle of odd length would contradict the property of being bipartite.

In summary, a bipartite graph is a graph in which the vertices can be divided into two disjoint sets such that all edges connect vertices from different sets. Additionally, bipartite graphs do not contain cycles of odd length.